Realizations ofU q(su(1, 1)) by deformed quantum harmonic oscillator algebras and associated deformed Heisenberg algebras

Forsu(1, 1)-symmetric Hamiltonians of quantum mechanical systems (e.g. single-mode quantum harmonic oscillator, radial Schrödinger equation for Coulomb problem or isotropic quantum harmonic oscillator, etc.), the Heisenberg algebra of phase-space variables in two dimensions satisfy the bilinear commutation relation [ip,x]=1 (in normal units). Also there are different realizations ofsu(1, 1) by the generators of quantum harmonic oscillator algebra. We seek here the forms of deformed Heisenberg algebras (bilinear in deformedx and ip) associated with deformedsu(1, 1)-symmetric Hamiltonians. These forms are not unique in contrast to the undeformed case; and these forms are obtained here by considering different realizations of the deformedsu(1, 1) algebra by deformed oscillator algebras (satisfying different bilinear relations in deformed creation and annihilation operators), and then imposing different conditions (e.g. the deformed Heisenberg algebra of the form of the undeformed one, the form of realizations of the deformedsu(1, 1) algebra by deformed phase-space variables being the same as that ofsu(1, 1) algebra by undeformed phase-space variables, etc.), assuming linear relations between deformed phase-space variables and deformed creation-annihilation operators (as it is done in the undeformed case), we get different Heisenberg algebras. These facts are revealed in the case of a two-body Calogero model in its centre of mass frame (and for no other integrable systems in one-dimension having potential of the formV(x _i ? x_j).     

Some comments on q-deformed oscillators and q-deformed su(2) algebras

The various relations between q-deformed oscillators algebras and the q-deformed su(2) algebras are discussed. In particular, we exhibit the similarity of the q-deformed su(2) algebra obtained from q- oscillators via Schwinger construction and those obtained from q-Holstein-Primakoff transformation and show how the relation between $su_{\sqrt q } (2)$ and Hong Yan q-oscillator can be regarded as an special case of Inöuë- Wigner contraction. This latter observation and the imposition of positive norm requirement suggest that Hong Yan q-oscillator algebra is different from the usual $su_{\sqrt q } (2)$ algebra, contrary to current belief in the literature.     

The classical su(2) invariance of the su(2) q -invariant XXZ spin chain

The XXZ spin-chain Hamiltonian has been constructed to be su(2) _q -invariant, but naively does not appear to be su(2)-invariant. However, using recently discovered deforming maps between representations of su(2) _q and corresponding representations of su(2), we prove a theorem which states that if a Hamiltonian is su(2) _q -invariant, it is also su(2)-invariant. The theorem generalizes to any quantized Lie algebra.     

Non-periodic Ishibashi states: the su(2) and su(3) affine theories

We consider the su(2) and su(3) affine theories on a cylinder, from the point of view of their discrete internal symmetries. To this end, we adapt the usual treatment of boundary conditions leading to the Cardy equation to take the symmetry group into account. In this context, the role of the Ishibashi states from all (non-periodic) bulk sectors is emphasized. This formalism is then applied to the su(2) and su(3) models, for which we determine the action of the symmetry group on the boundary conditions, and we compute the twisted partition functions. Most if not all data relevant to the symmetry properties of a specific model are hidden in the graphs associated with its partition function, and their subgraphs. A synoptic table is provided that summarizes the many connections between the graphs and the symmetry data that are to be expected in general.     

su q (2)-invariant harmonic oscillator

We propose a q-deformation of the su(2)-invariant Schrödinger equation of a spinless particle in a central potential, which allows us not only to determine a deformed spectrum and the corresponding eigenstates, as in other approaches, but also to calculate the expectation values of some physically-relevant operators. Here we consider the case of the isotropic harmonic oscillator and of the quadrupole operator governing its interaction with an external field. We obtain the spectrum and wave functions both for q R⁺ and generic q S ¹, and study the effects of the q-value range and of the arbitrariness in the su _q (2) Casimir operator choice. We then show that the quadrupole operator in l = 0 states provides a good measure of the deformation influence on the wave functions and on the Hilbert space spanned by them.     

Contraction of graded su(2) algebra

The Inönu-Wigner contraction scheme is extended to Lie superalgebras. The structure and representations of extended BRS algebra are obtained from contraction of the graded su(2) algebra. From cohomological consideration, we demonstrate that the graded su(2) algebra is the only superalgebra which, on contraction, yields the full BRS algebra.     

A note on open‐chain transfer matrices from q‐deformed su(2|2)S‐matrices

In this note, we perform Sklyanin's construction of commuting open‐chain/boundary transfer matrices to the q‐deformed SU(2|2) bulk S‐matrix of Beisert and Koroteev and a corresponding boundary S‐matrix. This also includes a corresponding commuting transfer matrix using the graded version of the q‐deformed bulk S‐matrix. Utilizing the crossing property for the bulk S‐matrix, we argue that the transfer matrix for both graded and non‐graded versions contains a crucial factor which is essential for commutativity.     

Extremal vectors for Verma-type representations of su(2, 2)

Starting from the Verma modules of the algebra sl(4, ?) we explicitly construct factor representations of the algebra su(2, 2) which are connected with unitary representation of group SU(2, 2). We find a full set of extremal vectors for this kind of representations, so we can solve explicitly the problem of irreducibility of these representations.     

The Baker-Campbell-Hausdorff formula for the chiral su(2) supergroup

In the theory of supersymmetricSU(2) Yang-Mills fields described on the 8th dimensional superspace, the local gauge transformations constitute a group whose Lie algebra has as coefficients belonging to the Weyl-spinorial Grassmann algebra.We present here a Baker-Campbell-Hausdorff formula for the chiralSU(2) supergroup and using this formula we give the finite form of each element of this group in terms of the local fields emering in the infinitesimal real superscalar generator.On leave at Departamento de Física, Universidad Simón Bolivar, Apartado 80659, Caracas 108 Venezuela     

The su(n) Hubbard model

The one-dimensional Hubbard model is known to possess an extended su(2) symmetry and to be integrable. I introduce an integrable model with an extended su(n) symmetry. This model contains the usual su(2) Hubbard model and has a set of features that makes it the natural su(n) generalization of the Hubbard model. Complete integrability is shown by introducing the L-matrix and showing that the transfer matrix commutes with the Hamiltonian. While the model is integrable in one dimension, it provides a generalization of the Hubbard Hamiltonian in any dimension.     

CALCULATION OF THE FIXED LENGTH FUNDAMENTAL su(2) HIGGS MODEL

The phase diagram of the lattice system of SU(2) gauge field coupled with the fixed length Higgs field in fundamental representation has been calculated by the variational-cumulant expansion method to the third order approximation.The method of determining the variational parameter has been improved by using the free energy to the second order approximation.Thus calculated phase diagram is in good agreement with the Monte Carlo estimation and the order of the phase transition is clearly determined in the third order approximation.     

The representations and coupling coefficients of su(n); application to su(4)

Analytical expressions for the matrices and an explicit algorithm for computing Clebsch-Gordan coupling coefficients are given forsu(4) in au(3)-coupled basis as an example of the construction for anysu(n) in au(n−1) basis. The results areinduced from the known results foru(3) by means of the vector-coherent-state (VCS) theory of induced representations. The important recent result that makes this possible is the discovery that a complete set of shift tensors for the finitedimensional representations of reductive Lie algebras can be induced, by VCS methods, from those of suitably defined subalgebras.     

THE WILSON LOOP EXPECTATION VALUES IN 2-AND 3-DIMENSIONAL su(2) LGT

An improved Monte Carlo scheme is applied to the computation of the expectation values of nxm Wilson loops in both 2-and 3-dimensional SU(2) lattice gauge theories.The results are compared with those simulated by the discrete group Y₁₂₀ and the exact results in two dimensions.     

Schrödinger equations for su q (2)-invariant quantum systems

By deforming the Hamiltonian of a spinless particle in a central potential we set up su _q (2)-invariant Schrödinger equations within the usual framework of quantum mechanics. Different deformations correspond to a given Hamiltonian. We explicitly solve different stationary Schrödinger equations for the free particle and for the hydrogen atom, and compare the associated energy spectra.     

Lie group exponents andSU(2) current algebras

Due to the Cappelli-Itzykson-Zuber classification, the minimal conformally invariant quantum field theories withSU(2) currents are classified by the ADE Lie algebras. Here I give a conceptual proof of the empirically valid relation between their partition functions and the Lie algebra exponents.